1. ECEN 460 Power System Operation and Control
Lecture 11: Power flow solution methods
Adam Birchfield
Dept. of Electrical and Computer Engineering
Texas A&M University
Material gratefully adapted with permission from slides by Prof. Tom Overbye.

2. Power balance equations

3. Power balance equations, cont’d

4. Real power balance equations

5. Slack bus
We can not arbitrarily specify S at all buses because total generation must equal total load + total losses
We also need an angle reference bus.
To solve these problems we define one bus as the "slack" bus (also known as the “swing bus”). This bus has a fixed voltage magnitude and angle, and a varying real/reactive power injection.
Picks up the slack in real and reactive power
Predefined known voltage angle, such as 0°

6. Why a slack bus is needed
Consider a three bus system with the specified transmission line impedances
This Ybus is actually singular!
So we cannot solve
This means (as you might expect), we cannot independently specify all the current injections I

7. Announcements Exam 1 will be Tuesday October 9 Please read Chapter 6
Closed-book, closed-notes, regular calculator and one 8.5”x11” notesheet are allowed
No lab Oct 5-9 due to exam
Drop-in office hours Friday Oct. 5 from 8 am to 12 pm in Zach 252
Please read Chapter 6
Quizzes most Thursdays
This week 10/4 it will be on per-unit
Next week 10/11 it will be on building a Y-bus

8. Nonlinear systems may have multiple solutions or no solution
Example 1: x2 - 2 = 0 has solutions x = 1.414…
Example 2: x2 + 2 = 0 has no real solution
f(x) = x2 - 2
f(x) = x2 + 2
two solutions where f(x) = 0
no solution f(x) = 0

9. Multiple solution example 3
The dc system shown below has two solutions:
where the 18 watt
load is a resistive
load
What is the
maximum
PLoad?

10. Power flow requires iterative solution

11. Gauss iteration

12. Gauss iteration example

13. Stopping criteria

14. Gauss power flow

15. Gauss two bus power flow example
A 100 MW, 50 Mvar load is connected to a generator through a line with z = j0.06 p.u. and line charging of 5 Mvar on each end (100 MVA base). Also, there is a 25 Mvar capacitor at bus 2. If the generator voltage is 1.0 p.u., what is V2?
SLoad = j0.5 p.u.

16. Gauss two bus example, cont’d

17. Gauss two bus example, cont’d

18. Gauss two bus example, cont’d

19. Gauss with many bus systems

20. Gauss-Seidel iteration

21. Three types of power flow buses
There are three main types of power flow buses
Load (PQ) at which P/Q are fixed; iteration solves for voltage magnitude and angle.
Slack at which the voltage magnitude and angle are fixed; iteration solves for P/Q injections
Generator (PV) at which P and |V| are fixed; iteration solves for voltage angle and Q injection
special coding is needed to include PV buses in the Gauss-Seidel iteration (covered in book, but not in slides since Gauss-Seidel is no longer commonly used)

22. Accelerated G-S convergence

23. Accelerated convergence, cont’d

24. Gauss-Seidel advantages and disadvantages
Each iteration is relatively fast (computational order is proportional to number of branches + number of buses in the system
Relatively easy to program
Disadvantages
Tends to converge relatively slowly, although this can be improved with acceleration
Has tendency to miss solutions, particularly on large systems
Tends to diverge on cases with negative branch reactances (common with compensated lines)
Need to program using complex numbers